Homogeneous polynomials and spurious local minima on the unit sphere

TL;DR

This paper characterizes all local minima of generic homogeneous polynomials on the unit sphere using simple criteria involving function value, gradient norm, and Hessian eigenvalues, and identifies conditions for absence of spurious minima.

Contribution

It provides a compact, algebraic-geometry-based characterization of local minima and conditions for no spurious minima for generic degree-d forms on the sphere.

Findings

All local minima can be characterized by simple spectral and value criteria.

A polynomial Res determines when all critical points are local minima.

Conditions for no spurious local minima are established using gradient ideals.

Abstract

We consider degree-d forms on the Euclidean unit sphere. We specialize to our setting a genericity result by Nie obtained in a more general framework. We exhibit an homogeneous polynomial Res in the coefficients of f , such that if Res(f) = 0 then all points that satisfy first-and second-order necessary optimality conditions are in fact local minima of f on the unit sphere. Then we obtain obtain a simple and compact characterization of all local minima of generic degree-d forms, solely in terms of the value of (i) f , (ii) the norm of its gradient, and (iii) the first two smallest eigenvalues of its Hessian, all evaluated at the point. In fact this property also holds for twice continuous differentiable functions that are positively homogeneous. Finally we obtain a characterization of generic degree-d forms with no spurious local minimum on the unit sphere by using a property of gradient ideals in algebraic geometry.